Classical Electrodynamics · Problem 14.21

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bohrs-correspondence-principle-radiation

Problem

Bohr's correspondence principle states that in the limit of large quantum numbers the classical power radiated in the fundamental is equal to the product of the quantum energy (ℏω0\hbar\omega_0) and the reciprocal mean lifetime of the transition from principal quantum number nn to (n−1)(n-1). (a) Using nonrelativistic approximations, show that in a hydrogen-like atom the transition probability (reciprocal mean lifetime) for a transition from a circular orbit of principal quantum number nn to (n−1)(n-1) is given classically by 1τ=23e2ℏc(Ze2ℏc)4mc2ℏ1n5\frac{1}{\tau} = \frac{2}{3}\frac{e^2}{\hbar c}\left(\frac{Ze^2}{\hbar c}\right)^4\frac{mc^2}{\hbar}\frac{1}{n^5} (b) For hydrogen compare the classical value from part a with the correct quantum-mechanical results for the mean lives of the transitions 2p→1s2p\to 1s (1.6×10−91.6\times 10^{-9} s), 4f→3d4f\to 3d (7.3×10−87.3\times 10^{-8} s), 6h→5g6h\to 5g (6.1×10−76.1\times 10^{-7} s).

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